Problem: Determine how many solutions exist for the system of equations. ${18x-3y = -9}$ ${y = 6x+3}$
Convert both equations to slope-intercept form: ${18x-3y = -9}$ $18x{-18x} - 3y = -9{-18x}$ $-3y = -9-18x$ $y = 3+6x$ ${y = 6x+3}$ ${y = 6x+3}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 6x+3}$ ${y = 6x+3}$ Both equations have the same slope and the same y-intercept, which means the lines would completely overlap. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Since any solution of ${18x-3y = -9}$ is also a solution of ${y = 6x+3}$, there are infinitely many solutions.